Ari Laptev, Timo Weidl (2000)
Journées équations aux dérivées partielles
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We give a survey of results on the Lieb-Thirring inequalities for the eigenvalue moments of Schrödinger operators. In particular, we discuss the optimal values of the constants therein for higher dimensions. We elaborate on certain generalisations and some open problems as well.
Pierre Duclos, Markus Klein (1985)
Journées équations aux dérivées partielles
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Zhongwei Shen (1996)
Journées équations aux dérivées partielles
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Jean Dolbeault, Ari Laptev, Michael Loss (2008)
Journal of the European Mathematical Society
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Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one dimension. This allows us to improve on the known estimates of best constants in Lieb–Thirring inequalities for the sum of the negative eigenvalues for multidimensional Schrödinger operators.
Giuseppe Maria Coclite (2002)
Annales Polonici Mathematici
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We prove the existence of a sequence of radial solutions with negative energy of the Schrödinger-Maxwell equations under the action of a negative potential.
A. Martin (1976)
Recherche Coopérative sur Programme n°25
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Christian G. Simader (1978)
Mathematische Zeitschrift
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Hansson, Anders M. (2005)
International Journal of Mathematics and Mathematical Sciences
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Astaburuaga, María Angélica, Briet, Philippe, Bruneau, Vincent, Fernández, Claudio, Raikov, Georgi (2008)
Serdica Mathematical Journal
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We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable...
Joachim Stubbe (2010)
Journal of the European Mathematical Society
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We show that phase space bounds on the eigenvalues of Schr¨odinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In particular, we provide a new proof of sharp Lieb– Thirring inequalities.
Victor Ivrii (1997)
Journées équations aux dérivées partielles
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Jecko, Thierry (2005)
Mathematical Physics Electronic Journal [electronic only]
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Tuan Duong, Anh (2012)
Serdica Mathematical Journal
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2010 Mathematics Subject Classification: 81Q20 (35P25, 81V10). The purpose of this paper is to study the Schrödinger operator P(B,w) = (Dx-By^2+Dy^2+w^2x^2+V(x,y),(x,y) О R^2, with the magnetic field B large enough and the constant w № 0 is fixed and proportional to the strength of the electric field. Under certain assumptions on the potential V, we prove the existence of resonances near Landau levels as B®Ґ. Moreover, we show that the width of resonances is of size O(B^-Ґ). ...